proposition A.571 Invariant Hamiltonians descend with their flows

Rests on

• depends_on equation 24.5 eq:sym-hamiltonian-field ¶
• depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
  • depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
    • depends_on definition A.561 Coadjoint action and equivariance ¶
      • depends_on definition 4.21 Group ¶
        • depends_on definition 4.11 Inverse element ¶
        • depends_on definition 4.20 Monoid ¶
      • depends_on definition 14.2 Lie group ¶
      • depends_on definition 24.43 Momentum map ¶
        • depends_on definition 14.2 Lie group ¶ ↺
        • depends_on definition 24.14 Hamiltonian vector field ¶
        • depends_on definition 24.17 Symplectomorphism ¶
    • depends_on proposition A.565 Freeness makes every value regular ¶
      • depends_on lemma A.564 Dimension and double orthogonal ¶
        • depends_on definition 24.2 Symplectic vector space ¶
        • depends_on theorem 5.40 Rank–nullity ¶
        • proves proof app:A-long-proofs@proof-339 ¶
      • depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
        • depends_on definition 24.8 Symplectic manifold ¶
        • depends_on definition 24.43 Momentum map ¶ ↺
        • depends_on theorem 13.59 Regular value theorem ¶
        • proves proof ch:07-symplectic-geometry@proof-15 ¶
      • depends_on theorem 13.59 Regular value theorem ¶ ↺
      • proves proof app:A-long-proofs@proof-340 ¶
    • depends_on theorem 13.67 Quotient manifold theorem ¶
      • depends_on definition 13.66 Smooth action; free; proper; orbit ¶
        • depends_on definition 13.48 Differentiable manifold ¶
        • depends_on definition 13.49 Smooth map between manifolds ¶
        • depends_on definition 6.9 Compact set ¶
      • depends_on definition 13.54 Embedded submanifold ¶
        • depends_on definition 13.53 Immersion, submersion, embedding ¶
        • depends_on definition 13.48 Differentiable manifold ¶ ↺
      • depends_on theorem 13.63 Constant rank theorem ¶
        • depends_on corollary A.292 Inverse function theorem ¶
        • depends_on definition 7.98 Functions of class $C^{1}$ ¶
        • depends_on theorem A.286 Implicit function theorem ¶
        • proves proof ch:11-manifolds-tensors-curvature@proof-12 ¶
      • proves proof app:A-long-proofs@proof-329 ¶
      • proves proof ch:11-manifolds-tensors-curvature@prooflink-2 ¶
    • proves proof app:A-long-proofs@proof-341 ¶
  • depends_on theorem A.568 The radical is the isotropy orbit ¶
    • depends_on lemma A.567 The differential of the momentum map along the orbit ¶
      • depends_on definition A.561 Coadjoint action and equivariance ¶ ↺
      • depends_on proposition A.565 Freeness makes every value regular ¶ ↺
      • depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶
        • depends_on definition 13.124 Integral curve; complete vector field ¶
        • depends_on theorem A.74 Flow of a time-dependent vector field ¶
        • depends_on theorem 9.8 Picard–Lindelöf ¶
        • proves proof ch:11-manifolds-tensors-curvature@proof-27 ¶
      • proves proof app:A-long-proofs@proof-342 ¶
    • depends_on lemma A.564 Dimension and double orthogonal ¶ ↺
    • depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶ ↺
    • proves proof app:A-long-proofs@proof-343 ¶
  • depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
    • depends_on definition 24.14 Hamiltonian vector field ¶ ↺
    • depends_on definition 24.8 Symplectic manifold ¶ ↺
    • depends_on proposition 13.128 Cartan's magic formula ¶
      • depends_on definition 13.103 Exterior derivative ¶
        • depends_on definition 13.98 $k$-form ¶
        • depends_on definition 13.102 Wedge product ¶
      • depends_on equation 13.246 eq:mfd-leibniz-forms ¶
      • depends_on equation 13.245 eq:mfd-nilpotency ¶
      • depends_on proposition 13.127 Component formulas ¶
        • depends_on definition 13.89 Contravariant and covariant tensors ¶
        • depends_on definition 13.90 Mixed tensor ¶
        • depends_on equation 13.267 eq:mfd-lie-def ¶
        • proves proof ch:11-manifolds-tensors-curvature@proof-28 ¶
      • proves proof ch:11-manifolds-tensors-curvature@proof-29 ¶
    • proves proof ch:07-symplectic-geometry@proof-5 ¶
  • proves proof app:A-long-proofs@proof-344 ¶
• depends_on theorem 24.45 Noether, symplectic form ¶
  • depends_on definition 24.43 Momentum map ¶ ↺
  • depends_on equation 24.5 eq:sym-hamiltonian-field ¶ ↺
  • depends_on proposition 24.15 Poisson bracket from the symplectic form ¶
    • depends_on definition 22.29 Poisson bracket ¶
      • depends_on definition 7.97 Partial derivative; gradient ¶
        • depends_on definition 7.26 Derivative of a function at a point ¶
        • depends_on definition 5.15 Basis ¶
      • depends_on definition 22.14 Phase-space coordinate transformation ¶
        • depends_on definition 22.2 Hamiltonian ¶
    • depends_on definition 24.14 Hamiltonian vector field ¶ ↺
    • depends_on equation 22.46 eq:ham-jacobi ¶
    • proves proof ch:07-symplectic-geometry@proof-4 ¶
  • proves proof ch:07-symplectic-geometry@proof-14 ¶
• proves proof app:A-long-proofs@proof-345 ¶

Supports

Nothing declares a dependency on this node yet.